frbclv_wp1982-02.pdf

Working Paper 8202

STABILITY I N A MODEL OF STAGGERED-RESERVE ACCOUNTING

by Michael L. Bagshaw and W i l l i a m T. Gavin

The au thors a r e g r a t e f u l t o Robert Avery, R ichard Davis, W i l l i a m Dewald, Robert Laurent, David Lindsey, E.J. Stevens, and members o f t h e research s ta f f of t h e Federa l Reserve Bank o f Cleveland f o r h e l p f u l comments and suggest ions.

Working papers o f t h Bank o f Cleveland a re p r e l i m i n a r y ma te r i a l s , c i r c u l a t e d t o s t i m u l a t e d i scuss ion and c r i t i c a l comment. The views expressed h e r e i n a re those o f t h e authors and n o t n e c e s s a r i l y those of t h e Federa l Reserve Bank o f Cleveland o r t h e Board o f Governors o f t h e Federa l R

Rev i sed Federa l

August 1982 Reserve Bank o f Cleveland

http://clevelandfed.org/research/workpaper/index.cfmBest available copy

Stability in a Model of Staggered-Reserve Accounting

Abstract

Critics of staggered-reserve accounting have used simple models to

show that a disturbance to deposits with no change in total reserves sets

in motion an undamped cycle in which deposits oscillate above and below

the equilibrium implied by the total reserve target. In this paper a

simple reduced-form model of the money-supply process is used to

investigate the nature of the dynamic process implied by

staggered-reserve accounting. The parameters in the model include the

number of banking groups in the staggered regime, the reserve

requirement, the response of banks to their own reserve position, and the

response of banks to a deviation of the money supply from target.

Classical stability algorithms are used to range of

parameters for which the model is stable. In this paper, the model is

defined to be stable if the reduced-form difference equation for the

money supply represents a converging process.

The results confirm the presence of a perpetual cycle found by

others. This perpetual cycle depends on two special c

is that there are only two groups of banks in the sta

the second is that banks ignore information about the money supply and

Federal Reserve policy in making their asset portfolio decisions. When

del is extended to include more than two banking groups, or when

s are allowed to react to aggregate information, the money supply

rges to the target level following a disturbance to equilibrium.


I. I n t r o d u c t i o n

I n t h i s paper we p resen t a genera l model of t h e money-supply process

w i t h s taggered- reserve account ing. ' The fundamental b u i l d i n g b l ock i s

t he m u l t i p l i e r r e l a t i o n s h i p between t o t a l reserves and demand depos i ts .

We a b s t r a c t f r om t h e problems assoc ia ted w i t h lagged rese rve account ing,

t he l e n g t h o f t h e se t t lement per iod, d i f f e r e n t t ypes o f depos i ts ,

d i f f e r e n t i a l r ese rve requirements, t h e demand f o r currency, t h e demand

f o r excess reserves, and t h e u n c o n t r o l l a b l e f a c t o r s a f f e c t i n g reserve

supply. It i s assumed t h a t t h e Federa l Reserve System s e t s t h e l e v e l o f

t o t a l reserves, c l oses t h e d i scoun t window, and a l l ows no car ryover .

The purpose o f t h i s model i s t o analyze a s s e r t i o n s about t h e dynamic

response o f depos i t s t o a d is tu rbance of t h e money supply f rom t h e t a r g e t

l e v e l . Th i s t a r g e t l e v e l i s assumed t o be t h e e q u i l i b r i u m l e v e l i n a more

complete u n s p e c i f i e d model o f t h e money-supply process and t h e economy.

L indsey (1981) s t a t e s t h a t a n a l y s i s done a t t h e Federal Reserve Board on s taggered- reserve account ing suggests t h a t dynamic i n s t a b i l i t i e s may be

i nhe ren t i n staggered account ing pe r se.' We f i n d , as Trepe ta and

L indsey ( 1 979) have found, t h a t under reasonable assumptions about banks' r e a c t i o n s t o i n d i v i d u a l r ese rve shortages ( o r surp luses) , random shocks t o t h e money s tock induce undamped c y c l e s i n t h e aggregate l e v e l o f

depos i ts . However, t h e f a i l u r e o f depos i t s t o converge t o t h e e q u i l i b r i u m

l e v e l seems improbable because a c y c l e i n depos i t s would t end t o induce a

c y c l e i n t h e f e d e r a l funds r a t e and i m p l y a p r o f i t o p p o r t u n i t y t h a t would

be easy f o r banks t o e x p l o i t . I n t h i s paper we show tha t , even i f banks

i g n o r e t h i s p r o f i t oppor tun i t y , t h e dynamic i n s t a b i l i t y descr ibed i n t h e

Trepeta- Lindsey paper i s p e c u l i a r t o a model w i t h j u s t two banking


groups. When t h e model i s extended t o i n c l u d e more than two banking

groups, t h e dynamic i n s t a b i l i t y disappears.

11. The Model

The model t h a t i s used t o examine t h e assumptions about t h e

i n s t i t u t i o n a l s t r u c t u r e and economic behav io r t h a t a re l i k e l y t o produce

dynamic i n s t a b i l i t y i n t h e money-supply process i s shown i n t a b l e 1. The

banks a re d i v i d e d i n t o n homogeneous groups, w i t h one group s e t t l i n g i n

each o f n success ive per iods. Required reserves a re c a l c u l a t e d on a

contemporaneous b a s i s over t h e n per iods. Banks t h a t a re s h o r t o f

reserves on se t t l emen t day must borrow reserves f r om o t h e r banks. I n t h e

aggregate, i f t h e members o f a s e t t l i n g group a r e sho r t o f reserves, t h e y

must be n e t borrowers f r om t h e n o n s e t t l i n g g roup(s ) . I t i s assumed t h a t depos i t s a re s p l i t even ly among banking groups. The f i r s t equa t ion i s a

behav io ra l equa t i on i n which t h e money supp ly changes i n response t o bank

behavior . Changes i n t h e money supp ly a l s o depend on t h e nonbank p u b l i c ' s

behavior , b u t t h a t behav io r i s ignored so t h a t we may focus on t h e

" c y c l i n g t t phenomena repo r ted by Laufenberg (1975) and Trepeta and L indsey ( 1979).

The f i r s t equa t ion descr ibes aggregate behav io r based on a model o f a

s i n g l e bank 's behavior . The equat ion con ta ins two behav io ra l parameters,

p and d; p measures t h e r e a c t i o n o f an i n d i v i d u a l bank t o a d e v i a t i o n

ween ac tua l and r e q u i r e d reserves. I f t h e money supply goes above t h e

t l e v e l , t h e demand f o r reserves w i l l exceed t h e supply and each

e t t l i n g bank w i l l borrow a p ropo r t i on , p, o f i t s d e f i c i t , caus ing

e s t r a t e s t o r i s e . As each bank a d j u s t s t h e asset s i d e o f i t s

ce sheet, depos i t s w i l l t end t o f a l l back t o t h e t a r g e t l e v e l .


Table 1

Model o f a Staggered-Reserve Account ing Regime

(3 . ) ARNt = ( T R - ARSt)/(n - 1).

M = t h e money supp ly

TK = t o t a l reserves

RRN = r e q u i r e d reserves o f a t y p i c a l non- se t t

ARN = a c t u a l r a t y p i c a l n o n- s e t t l i

ARS = ac tua l

a t i o n o f t h e money supply f r om


Since t o t a l reserves a r e f i x e d , banks w i l l n o t be a b l e t o ach ieve t h e

p o r t f o l i o mix d e s i r e d a t t h e i n i t i a l l e v e l of i n t e r e s t r a tes . Therefore,

p r i c e s o f o t h e r asse ts w i l l change u n t i l t h e i n d i v i d u a l banks a re

s a t i s f i e d t o h o l d a v a i l a b l e reserves. The s e t t l i n g banks w i l l have t o

borrow any d e f i c i t (and a re assumed t o l end a l l excess reserves) on se t t l emen t day. The g r e a t e r t h e s i z e o f t h e r e s e r v e imbalance, t h e more

i n t e r e s t r a t e s w i l l have t o change. The a c t i o n s o f t h e s e t t l i n g banks

w i l l a f f e c t t h e money supp ly i n t h e same way as t h e a c t i o n s o f t h e

n o n- s e t t l i n g banks. Therefore, t h e r ese rve d e f i c i t o r su rp l us o f a

t y p i c a l n o n - s e t t l i n g bank ing group i s m u l t i p l i e d b y t h e number o f bank ing 3 groups. I f p = 1, then t h e t o t a l depos i t e f f e c t o f t h e i n d i v i d u a l bank

r e a c t i o n s t o t h e i r own rese rve p o s i t i o n s w i l l be equal t o n t imes t h e

d i f f e r e n c e between t h e r e q u i r e d and a c t u a l reserves o f a n o n - s e t t l i n g

bank ing group. If t h e rese rve requi rement i s l e s s t han 100 percent , t h i s

e f f e c t i s l e s s t han t h e amount by which t o t a l d e p o s i t s would have t o

change t o r e t u r n t h e t o t h e t a r g e t l e v e l

The o t h e r behavo r i a l parameter i s d. It measures aggregate bank

r e a c t i o n t o a d e v i a t ney supp ly from t h e t a r g e t . Equa t ion 1

i nco rpo ra tes r e a c t i o es o f i n f o rma t i on . The f i r s t i s i n t e r n a l

and represen ts an i n d i v i a u a l bank 's r e a c t i o n t o i

pos i t i on. The second i r e a c t i o n t o aggregate on a v a i l a b l e

t o a l l banks. I n a d d i t i o n , t h i s equa t ion i n c l u d e s a d i s t u rbance t e rm t h a t

r ep resen t s shocks t h a t o r i g i n a t e f r om o u t s i d e t h i s model.

Equa t ion 2 i s 1 equat ion.

p t i o n t h a t t h e s e t t l i n g banks w i l l never

they w i l l always meet t h e i r r ese rve requ

I t i s

h o l d e

i rement s.

and


Equat ion 3 i s a d e f i n i t i o n used t o c a l c u l a t e ac tua l reserves o f a

t y p i c a l group o f n o n - s e t t l i n g banks. The d i s c o u n t window i s c losed, and

t h e r e i s no c a r r y o v e r so t h a t ac tua l reserves o f t h e s e t t l i n g banks w i l l

equal r e q u i r e d reserves. Ac tua l reserves o u t s i d e t h e s e t t l i n g banks a re

assumed t o be d i v i d e d even ly among t h e n o n - s e t t l i n g banks.

The genera l s o l u t i o n f o r t h e dynamic p a t h o f t h e money supply i n

response t o exogenous shocks i s g i ven below. No t i ng t h a t RRNt = q/n

Mt and s u b s t i t u t i n g 2 i n t o 3, t h e model reduces t o equat ions 1 and 3 ' :

Using l a g ope ra to r s and s u b s t i t u t i n g equa t i on 3 ' i n t o 1, we ge t

t h o f Mt w i l l be determined by a

combinat ion o f t h e r e

response t o

s o l u t i o n of

1 i s s t a b l e c r n = 2, 3, o r 4.


The d i f f e r e n c e equat ion de r i ved from equa t i on 4 under t h e assumption

t h a t d = 0 i s g i ven by

Th i s process w i l l converge i f a l l t h e r o o t s o f t h e po lynomia l ,

n - 1 - ( n - npq)B + Bn, l i e o u t s i d e t h e u n i t c i r c l e . I f any r o o t l i e s on o r w i t h i n t h e u n i t c i r c l e , t h e process w i l l n o t converge.

For t h e case i n which t h e r e a r e two banking groups, t h e po lynomia l i n

t h e denominator o f equa t ion 5 i s

The r o o t s o f t h e po lynomia l a r e g iven by

For pq equal t o 0, bo th r o o t s equal 1; f o r pq equal t o 2, bo th r o o t s

equal -1. For pq i n s i d e t h e range 0 t o 2, t h e r o o t s a re complex; t h e

d i s tance o f t h e r o o t s f r om t h e o r i g i n i s g i ven by

I n t h i s spec ia l case of two banking groups w i t h d = 0, 0 r pq I 2, b o t h

r o o t s l i e on t h e u n i t c i r c l e , and t h e p a t h f o r t h e money supply f o l l o w i n g

an exogenous shoc cyc 1 e.

A p r i o r i , - be i n t h i s rang ve

requirement, q, i s o f 10 percent . The parameter, p,

d e s c r i b i n g t h e ban r v e imbalances should be i n t h e

reason f o r a b

u n l i k e l y i f ban


I f t h e p roduc t pq i s g r e a t e r than 2 o r l e s s than 0, b o t h r o o t s w i l l

be r e a l . One r o o t w i l l be l e s s than 1 i n abso lu te value; t h e o t h e r r o o t

w i l l be g r e a t e r than 1 i n abso lu te value, and t h e dynamic process

descr ibed i n equa t ion 5 explodes. The economic i n t e r p r e t a t i o n i s

s t r a i g h t f o r w a r d . I f pq< 0, then banks a r e p e r s i s t e n t l y l end ing a t an

i n t e r e s t r a t e t h a t i s below t h e expected i n t e r e s t r a t e i n t h e nex t

per iod, o r t h e y a re p e r s i s t e n t l y borrowing a t an i n t e r e s t r a t e t h a t i s

above t h e expected i n t e r e s t r a t e i n t h e n e x t per iod . I f pq i s g r e a t e r

than 2, and q i s i n t h e neighborhood o f 10 percent , then banks a re

g r e a t l y o v e r r e a c t i n g t o any rese rve imbalance.

To analyze t h e cases when n i s g r e a t e r t han 2, no te t h a t t h e

po lynomia l i n equa t i on 6 has r o o t s t h a t a r e t h e i nve rse o f t h e r o o t s i n

equa t ion 7.

( 6 ) f (B ) = n-1 - n ( l - pq)B + B ~ .

D u f f i n (1969) presented an a l g o r i t h m f o r t e s t i n g whether a l l t h e r o o t s o f a po lynomia l l i e i n s i d e t h e u n i t c i r c l e . I f t h e r o o t s o f g(B) l i e i n s i d e t h e u n i t c i r c l e , then t h e r o o t s o f f ( B ) l i e o u t s i d e t h e u n i t c i r c l e , and equat ion 5 r ep resen ts a s t a b l e d i f f e r e n c e equat ion. The d e t a i l s o f t h i s

a l g o r i t h m are presented i n appendix A w i t h t h e s o l u t i o n s f o r t h e cases

n = 3 and n = 4. Eq i s s t a b l e f o r t h e f o l l o w i n g va lues o f p and q:

Case 1: n =

Case 2: n = 4,


For t h e genera l case o f n banking groups, a n a l y t i c c a l c u l a t i o n o f

bounds on pq f o r convergence proved t o be i n t r a c t a b l e . However, f o l l o w i n g

computer s o l u t i o n s o f t h e model f o r cases n equals 5 through 30 and a

l a r g e v a r i e t y of values f o r pq, we hypothesize t h a t t h e dynamic process

descr ibed i n equat ion 5 converges f o r t h e f o l l o w i n g values o f pq:

Case 1: n > 2 and odd,

o < pq < 2(*). Case 2: n > 2 and even,

o < pq < 2.

I V . S t a b i l i t y Condi t ions w i t h a React ion t o Aggregate In fo rma t i on

I n t h i s s e c t i o n we r e l a x t h e assumption t h a t d = 0. The d i f f e r e n c e

equat ion der ived f rom equat ion 4 i n t h e general case i s g iven by n-1

(n - 1 - ZBi)et (8) Mt = l / q TR - i =l

i n - 1 - n(1-pq-d)B + B~ - d 2 B

i = l hhen n = 2, equat ion 8 i s s t a b l e i f and o n l y i f bo th r o o t s o f t h e

polynomi a1

( 9 ) 1 - [ 2 ( 1 - p q ) - d ) B + ( l - d ) B 2 l i e ou ts ide t h e u n i t c t h e general case t h e procedure developed

by D u f f i n and used above i s i n t r a c t a b l e f o r n > 2. I n use a

method developed by Wise (1956) t o d e r i v e s t a b i l i t y condi t ions. For n =2 , hod y i e l d s t h e f o l l o w i n g cond i t i ons (see appendix B f o r


It i s impor tan t t o no te t h a t w h i l e t h e system w i t h two banking groups d i d

no t converge f o r any values o f pq when d = 0, i t converges f o r a l l l i k e l y

values o f p and q when d l i e s between 0 and 2. The g rea t a t t e n t i o n

attached t o t h e weekly re lease o f aggregate i n f o r m a t i o n on t h e money

supply suggests t h a t d i s l i k e l y t o be g r e a t e r than 0. While t h e ac tua l

va lue of a can o n l y be est imated, we know t h a t banks and t h e p u b l i c r e a c t

t o aggregate i n f o r m a t i o n i n t oday ' s market. Unless one would p r e d i c t a

d r a s t i c change i n behavior f o l l o w i n g t h e adopt ion o f staggered- reserve

accounting, d i s l i k e l y t o f a l l i n t h e s t a b l e range.

Appendix B a l so descr ibes t h e a lgo r i t hm used t o compute t h e s o l u t i o n s

t o equat ion 8 f o r h ighe r values o f n. Numeric s o l u t i o n s up t o t h e case

n = 30 i n d i c a t e t h e f o l l o w i n g s t a b i l i t y c o n d i t i o n s when d > 0.

Case 1: n > 2 and odd, n- 1 O < pq < y-(2 - d).

Case 2: n > 2 and even,

For d < 0 and n > 2, t h e r e i s a s t a b l e r e g i o n f o r some values o f pq. The

cases f o r n = 3, 4, and 5 are shown i n appendix B and p l o t t e d i n f i g u r e


V. Conclusion

The dynamic process descr ibed by equa t ion 5 i n c l u d e s no sys temat ic

r e a c t i o n i n t h e market t o aggregate in fo rmat ion . When t h e r e a re o n l y two

banking groups w i t h staggered reserve- se t t lement per iods , we f i n d t h a t

t h i s process rep resen ts an undamped, unexp los ive c y c l e f o r l i k e l y va lues

o f p, t h e bank p o r t f o l i o response, ana q, t h e rese rve requirement. When

t h e number o f bank ing groups i s inc reased t o 3 o r 4, and f o r l i k e l y

va lues o f p and q, t h e money supp ly converges i n a damped c y c l e t o t h e

t a r g e t l e v e l .

The dynamic process descr ibed i n equa t ion 8 ~ n c l u d e s a market

response t o aggregate i n fo rma t i on . If market p a r t i c i p a n t s respond t o a

d e v i a t i o n o f t h e money supp ly f r o m t a r g e t by changing i n t e r e s t r a t e s i n

t h e d i r e c t i o n o f t h e dev ia t i on , t hen t h e dynamic process descr ibed i n

equa t ion 8 i s l i k e l y t o converge even when t h e r e a re o n l y two banking

groups.


Footnotes

1. The proposal t o s tagger r ese rve account ing was f i r s t made by Cox and

Leach (1964); a l s o see comments f o l l o w i n g t h e a r t i c l e by S t e r n l i g h t (1964). T h i s proposal was r e s u r r e c t e d by t h e Morgan Guaranty Company (1981) and d iscussed by Gavin (1982).

2. These r e s u l t s come f rom t h e a n a l y s i s o f a model cons t ruc ted by

Trepeta; see t h e appendix t o Trepeta and L indsey (1979). The fundamental d i f f e r e n c e between t h e model presented here and Trepeta

and L indsey ' s i s t h a t we gene ra l i ze t o i n c l u d e more than two banking

groups, and we add a behav io ra l parameter t o cap tu re r e a c t i o n t o

aggregate i n fo rma t i on . Laufenberg (1975) f i r s t noted t h a t t h e i n s t i t u t i o n a l s t r u c t u r e o f s taggered- reserve account ing i m p l i e d t h e

p o s s i b i l i t y o f dynamic i n s t a b i l i t i e s .

3. Robert Avery, Federal Reserve Board s t a f f , has po in ted ou t t h a t when

n = 2, and t h e model i s changed so t h a t

- d(Mtml + l / q TR) - et, t h e range o f va lues o f pq f o r which t h e model i s s t a b l e i s reduced by

a f a c t o r of one-half . Changing t h e model i n t h i s way does no t change

any o f our q u a l i t a t i v e r e s u l t s .


Appendix A

S t a b i l i t y i n t h e Model w i t h d = 0

A Test f o r Schur Polynomials f rom D u f f i n (1969) A Schur polynomial i s de f i ned as a polynomial f o r which a l l t h e r o o t s

l i e i n s i d e t h e u n i t c i r c l e .

Algor i thm. Le t gn(w) be t h e polynomi a1 n gn(x) = a. + alx + a2x2 + . .. + anx ,

where

a. f 0, an t 0. L e t gn - , ( x ) be t h e reduced polynomial

o f degree n - 1. Then gn(x) i s a Schur polynomial i f and o n l y i f

( i i ) gn-] ( x ) i s a Schur polynomi a1 . Th is procedure i s i t e r a t e d n t imes t o ge t t h e parameter domain f o r which

gn(x) i s a Schur polynomial . Case: n = 3

g3(B) = 1 - 3 (1 - 3 B .

( i ) I l l < 121, ( i i ) g2(B) = 3 ( 1 - pq) - 6 ( 1 - pq)B + 3~ ' ,

i i ) '


2 2 (ii)' g2(B) = 48(1 - pq12 - 96(1 - pq)B + (64 - 16(1 - pq) )B . 2 (i ) ' 148 (1 - pq)'~ < 164 - 16(1 - pq) 1.

2 2 Both (1 - pq) and 4 - (1 - pq) are greater than 0 on the range

(ii)" g,(B) = [-(64 - 16(1 - pq)2)(96(1 - pq)) + (480 - ~q)~)(96(1 - pq))l + [(64 - 16(1 - pq)')' - (48(1 - pq)2)2]~.

1-(64 - 16(1 - pq1~)(96(1 - pq)) + 48(1 - pq)296(1 - pq)l <

for 0 < pq < 1 3 3(1 - pq) - 3(1 - pq ) > 0 and

2 - (1 - pq) - (1 - pq)4 > 0.

Therefore, A.1 - 2 - 3(1 - pq) - (1 - Pq)2 + 3(1 - Pq)3 - (1 - pq) 4

= f(l - pq) > 0. f(l - pq) = 0 when pq = 0

= 2 when pq = 1


To show t h a t f ( 1 - pq) > 0 f o r a l l 0 < pq < 1 n o t e t h a t 2 f ' ( 1 - pq) = 3 + 2 ( 1 - pq) - 9 ( 1 - pq) + 4 ( 1 - pq13

f ' ( 1 - pq) = 0 when pq = -.693, 0, and 1.443.

Therefore, f ' ( 1 - pq) > 0 f o r 0 < pq < 1, and f ( 1 - pq) > 0 f o r 0 < pq < 1.

For 1 < pq < 2

3 ( 1 - p q ) 3 - 3 ( 1 - pq) > 0 and

2 Therefore, A.1 - 2 + 3 ( 1 - pq) - ( 1 - pq) - 3 ( 1 - pq)3 - ( 1 - pq) 4

= h ( l - pq) > 0. h ( 1 - pq) = 2 when pq = 1,

= 0 when pq = 2.

To show t h a t h ( 1 - pq) > 0 f o r a l l 1 < pq < 2 n o t e t h a t 3 h ' ( 1 - pq) = -3 + 2 ( 1 - pq) + 9 ( 1 - pq)' + 4 ( 1 - pq) .

h ' ( 1 - pq) = 0 when pq = .557, 2, and 2.693. h ' ( 1 - 1.5) = -2.25.

Therefore, h ' ( l - pq) < 0 f o r 1 < pq < 2,and h ( 1 - pq) > 0 f o r 1 < pq < 2.

Therefore, equa t ion 5 represen ts a s t a b l e d i f f e r e n c e equat ion when n = 4 and

0 < pq < 2.


For the general case of n banking groups, ana ly t ic calcula t ion of bounds

on pq f o r convergence proved t o be in t rac tab le . However, following computer

solut ions of the model f o r cases n equals 5 through 30 and a large var ie ty of

values f o r pq, we hypothesize t h a t the dynamic process described in equation 5 \

converges f o r the following values of pq.

Case 1: n > 2 and odd,

Case 2 : n > 2 and even,


Appendix B

S t a b i l i t y o f t h e Model i n t h e General Case

The dynamic process descr ibed i n equat ion 8 w i l l be s t a b l e i f t h e

r o o t s o f

l i e ou ts ide t h e u n i t c i r c l e . Equ iva len t ly , t h e system w i l l be s t a b l e i f

t h e r o o t s o f

l i e i n s i d e t h e u n i t c i r c l e .

Wise (1956) presents a method o f t r ans fo rm ing equat ions o f t h e t ype B.2 such t h a t t h e cond i t i ons on r o o t s l y i n g i n s i d e t h e u n i t c i r c l e a re

equ i va len t t o t h e r e a l p a r t o f t h e r o o t s of

be ing l e s s than zero. I n 8.3 y and p, a re de f i ned as

C . i s t h e c o e f f i c i e n t o f yr i n t h e expansion o f r J

a1 = - n ( 1 - pq) - ( n - l ) d , n-1

- a 2 - . . a = - d and n-1 n'zi '

- 1 - d a n - -. n - 1


From t h i s p o i n t , we can use t h e Routh theorem (Chiang 1974, p. 546), which s t a t e s t h a t a po lynomia l o f t h e f o rm ( w i t h a. assumed g r e a t e r than 0)

n n-1 + 8.4 a. x + al x ... + an-l x + an = 0

has t h e r e a l p a r t s o f a l l i t s r o o t s nega t i ve i f and o n l y i f t h e f i r s t n

o f t h e f o l l o w i n g sequence o f determinants

a re a l l p o s i t i v e . I n app l y i ng t h i s theroem, i t should be remembered t h a t

lal I E al and t h a t we s e t am = 0 f o r m > n.

I n t h e p resen t problem, we have -

ai - Pn-i f o r i = 0, 1, ..., n.

Thus, t h e c o n d i t i o n s f o r s t a b i l i t y o f t h e system can t h e o r e t i c a l l y be

d e r i v e d f r om Wise's r e s u l t s and t h e Routh theorem. However, i n p r a c t i c e

t h e a n a l y t i c a l s o l u t i o n i s ex t reme ly compl icated. Therefore, we p resen t

t h e a n a l y t i c a l s o l u t i o n s f o r n = 2 and 3. Numeric s o l u t i o n s a re g i ven f o r

Case: n = 2 --

~XI = - [2 (1 -pq) -d l and

From Wise (1 956) :

1 + a 2 = 2 r 2 - Pq - d l ,

p 1 = 2 - 2a2 = 2d and

p2 = 1 + al + a2 = 2pq.


Thus, a. - - P2 = 2pq i s p o s i t i v e i f pq > 0. Otherwise,al l s igns o f

the pi 's must be reversed. I f pq > 0, then the cond i t io r ls o f the

Routh theorem are

la11 = PI > o o r equ iva len t l y , d > o and

2d[2 ( 2 - pq -d ) ] > 0. Since d > 0, we get

If pq < 0, then we have t o reverse s igns Defore app ly ing the Routh

a3

acl a2

theorem.

PI 0 '

"2 Po-

- -

Condi t ions are then

= p p > O o r 1 0

2d[2 ( 2 - pq -d ) ] > 0. Again, s ince d < 0

T h i s l a s t i n e q u a l i t y c o n t r a d i c t s pq < 0. Thus, t he re are no s t a b l e

s o l u t i o n s f o r which pq < 0. The

< pq < 2 -d and

> 0.

shown i n f i g u r e B.1.

s t a b l e reg ion i s g iven by


F i g u r e B . 1 Stab le Parameter Domain

Case: n = 3

a2 = - d and 2 ' = 1 - d a3 .

2

From Wise (1 956) :

Again, t n i s case d i v i d e s i n t o two subcases, depending on wnether p3

i s p o s i t i v e o r negat ive.

I f p j > 0 o r e q u i v a l e n t l y , pq > 0, ( a l l = p 2 > 0 - p q > - 26.


B.5 pq < d ( d - 3) f o r d > l and 1 - d

Pq > d ( d - 3) f o r d < 1. 1 - d

There i s no c o n s t r a i n t f rom t h i s c o n d i t i o n on pq when a = 0.

If po > 0, we get t h e same conditions as i n B.5 p lus p > 0 0

B.6 pq < 4 - 2a . 3

If p o < 0, we get t h e reverse of cond i t i ons g iven i n 8.5. Because we

cannot have both t h e cond i t i ons i n B.5 and t h e i r inverses, t he re i s no

s t a b l e s o l u t i o n when po < 0.

I f p3 < 0 o r e q u i v a l e n t l y pq < 0, we have

(which g ives t h e same c o n d i t i o n as B.5) and


I f po > 0, we g e t t h e r e v e r s e of c o n d i t i o n s i n B.5. Thus, t h e r e i s no

s t a b l e r e g i o n w i t h po > 0 and p3 < 0. F o r po < 0, we o b t a i n t h e

same c o n d i t i o n s as B.5 and

pq > 4 - 2d . 3

Because i t i s n o t p o s s i b l e t o have b o t h pq< -2d and pq > ( 4 - 2d)/3, t h e r e a re no s t a b l e c o n a i t i o n s f o r pq < 0.

I n summary, t h e S t a b l e r e g i o n f o r n = 3 i s g i ven by t h e s e t o f

c o n d i t i o n s ob ta ined f r om those g i v e n a b o v e by de te rmin ing which ones a r e

b i n d i n g c o n d i t i o n s . They a r e

Pq < 4 -2d, and -

3

pq > d ( d - 3) f o r d < 0. -

I - d


T h e o r e t i c a l l y , i t would De p o s s i ~ l e t o s o l v e f o r t h e s t a b i l i t y

r eg ions by us ing t h e above method f o r n > 3. However, t h e a p p l i c a t i o n i s

n o t p r a c t i c a l because o f t h e compl i ca ted express ions invo lved. Insteaa,

we have developea computer s o l u t i o n s f o r n up t o 30 us ing t h e above

method and an a l g o r i t h m developed by D u f f i n (1969) t o c a l c u l a t e t h e Crjls. From these s o l u t i o n s , we maKe t h e f o l l o w i n g hypothes is :

( 1 ) F o r d B O , t h e s t a b i l i t y r e g i o n i s g i v e n b y pq < n - - 1 ( 2 - d ) f o r n odd,

n

p q < 2 - d f o r n even.

( 2 ) F o r 3 < 0, t h e r e i s a s t a b i l i t y r e g i o n f o r which we have n o t been a b l e t o determine genera l fo rmu las f o r boundar ies except p a r t i a l l y f o r

t h e case where n i s odd. I n t h i s case, t h e upper boundary appears t o be

pq < n - - 1 ( 2 - d) . F i g u r e B.2 i l l u s t r a t e s s t a b i l i t y reg ions f o r cases i n n

which n = 3, 4, and 5.


F i g u r e B.2. S t a b l e Parameter Domains


- 2 6 -

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